Theorem 2.0.4 
Assume \(q, r\in C^\infty (\Omega )\) 
on some open set \(\Omega \subset \mathbb {R}^2\)  with \((0,0)\subset \Omega \) .
Let \(\mathcal {D}_{\underline{\mu }},\mathcal {D}_{\underline{\hat{\nu }}},\underline{\hat{\mu }}=\lbrace \hat{\mu }_1,\ldots ,\hat{\mu }_n\rbrace ,\underline{\hat{\nu }}=\lbrace \hat{\nu }_1,\ldots ,\hat{\nu }_n\rbrace ,\) 
be the pole and zero divisor of degree \(n,\)  respectively,
associated with \(q,r\)  and \(\phi (\cdot ,x,t_{\underline{r}})\in \mathcal {K}_n\) 
according to ([1]}, (4.16) and (4.17)),
\(\hat{\mu }_j(x,t_{\underline{r}})&=(\mu _j(x,t_{\underline{r}}),-\mu _j(x,t_{\underline{r}})^{2n_-}G_{\underline{n}}(\mu _j(x,t_{\underline{r}}),x,t_{\underline{r}})),\quad j=1,\ldots ,n,\\\hat{\nu }_j(x,t_{\underline{r}})&=(\nu _j(x,t_{\underline{r}}),\nu _j(x,t_{\underline{r}})^{2n_-}G_{\underline{n}}(\nu _j(x,t_{\underline{r}}),x,t_{\underline{r}})),\quad j=1,\ldots ,n.\) 
